Graphing Absolute Value Equations

Absolute Value Equation

A V-shaped graph that points upward or downward is the graph of an absolute value equation.

Translation A translation is a shift of a graph horizontally, vertically or diagonally (combination of vertical and horizontal translation).

Graphing Absolute Value Equations

Vertical TranslationThe graph of y = │x │ + k is a translation of y = │x │ .

if k is positive up k unitsif k is negative down k units

Horizontal TranslationThe graph of y = │x – h│ is a translation of y = │x │ .

if h is positive right h unitsif h is negative left h units

Graphing Absolute Value Equations

Diagonal Translation

A combination of vertical and horizontal translation.

The graph of y = │x – h│ + k is a translation of y = │x │.

Graphing Absolute Value Equations

Example 1: Graph the absolute value equation y = │x │- 5.

Solution: Step 1. Press y = key on you calculator. Step 2. Use the NUM feature of MATH screen on your

graphing calculator.Step 3. Choose 1: abs( feature on you calculator.Step 4. Enter the given equation.

X,Τ,θ,n , ) , - , 5

Step 5. Press GRAPH.

Cont (example 1)…

-6

-5

-4

-3

-2

-1

0

1

2

3

-8 -6 -4 -2 0 2 4 6 8

Do these…

Graph each function. Identify the vertex of each function.

1. y = │x │+ 2

2. y = │x - 4 │

3. y = │x - 6│-2

4. y = │x - 2│+ 1.

Answers:

1. 2.

3. 4.

Writing an Absolute Value Equation

Write an equation for each translation of y = │x│Example 1: 9 units up

9 units up is vertical translation so use y = │x│ + k

Since k is positive, the equation is y = │x│ + 9

Example 2: 2 units down2 units down is vertical translation so use

y = │x│ + kSince k is negative, the equation is

y = │x│ - 2

Writing an Absolute Value Equation

Write an equation for each translation of y = -│x│.Example 3: 5 units right

5 units right is horizontal translation so use y = -│x - h│

Since k is positive, the equation is y = -│x - 5│

Example 4: 3 units left3 units left is horizontal translation so use

y = -│x -h│Since k is negative, the equation is

y = -│x – (-3)│ y = -│x +3│

Writing an Absolute Value Equation

Write an equation for each translation of y = │x│.Example 5: 2 units up and 1 unit leftsince this involves horizontal and vertical translations use

y = │x - h│+ kSince k is positive and h is negative, the equation is

y = │x – (-1)│+ 2 y = │x +1│+ 2 Example 6: 5 units down and 4 units leftsince this involves horizontal and vertical translations use

y = │x - h│+ kSince k is negative and h is negative, the equation is

y = │x – (-4)│+ (-5) y = │x +4│ - 5

Do these…

Write an equation for each translation of the parent function y = x.

1. Left 9 units

2. Right 2 unit

3. Up 1 unit

4. Down 2/3 unit

5. Left 3 units and down 4 units

6. Right 5 units and up 1 unit

Answers: 1. y = │x + 9│ 2. y = │x - 2│ 3. y = │x │ + 1 4. y = │x │ - 2/3 5. y = │x +3│- 4 6. y = │x - 5│ + 1

Graph each absolute value equation then describe the translation of the parent function.

Example 1. y = x - 7 + 2

Answer: y = xis translated 7 units to the right and 2 units up.

Describing a Translation

Describing a Translation

Graph each absolute value equation then describe the translation of the parent function.

Example 2. y = -x + 3 - 1

Answer: y = -xis translated 3 units to the left and 1 unit down.

Do these…

Graph each absolute value equation then describe the translation of the parent function.

1. y = x + 1- 32. Y = x – 3 - 10

3. Y = x + 2 + 1

4. Y = - x - 1 - 65. Y = - x - 5 + 7

Write the equation of the given graph.

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9

Answer: Since the vertex is at the point (5, 3), then h= 5 and k = 3. Therefore, the equation is

y = │x – 5│+ 3.

Describe the translation from y = x+1- 2 to y = x - 3 + 4

Answer: 6 units up, 4 units right

Describe the translation from y = x – 3+ 1 to y = x - 1 - 2

Answer: 3 units down, 2 units left